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\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{复变函数练习2.1}
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\renewcommand{\today}{\number\year \,年 \number\month \,月 \number\day \,日}
\date{2024 年 3 月 18 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 01
设函数 $w=f(z)$ 在包含 $z_0$ 的区域 $D$ 内有定义。
\begin{enumerate}%[label={(\arabic*)}]
\item  什么时候称函数 $f(z)$ 在点 $z_0$ 可导？
\item  什么时候称函数 $f(z)$ 在点 $z_0$ 可微？
\item  什么时候称函数 $f(z)$ 在点 $z_0$ 解析？
%什么是 $w=f(z)$ 在点 $z_0$ 的微分？
\end{enumerate} 

\vspace{0.2cm}

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\item  %Problem 02
证明函数 $f(z)=\overline{z}$ 在复数平面上处处不可微。

\vspace{0.2cm}

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\item  %Problem 03
设函数 $f(z)=u(x,y)+iv(x,y)$ 在区域 $D$ 内有定义。证明 $f(z)$ 在 $D$ 内一点 $z=x+iy$ 可微的充要条件是下述两个条件都成立，
\begin{enumerate}%[label={(\arabic*)}]

\item  二元函数 $u(x,y), v(x,y)$ 在点 $(x,y)$ 可微。

\item  二元函数 $u(x,y), v(x,y)$ 在点 $(x,y)$ 在点 $(x,y)$ 满足柯西-黎曼方程：

$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}. $$ 

\end{enumerate} 

\vspace{0.2cm}

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\item  %Problem 04
讨论函数 $f(z)=|z|^2$ 的解析性。

\vspace{0.2cm}

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\item  %Problem 05
讨论函数 $f(z)=x^2-iy$ 的可微性和解析性。

\vspace{0.2cm}

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\item  %Problem 06
设 $f(z)=x^2+axy+by^2+i(cx^2+dxy+y^2)$, 问常数 $a,b,c,d$ 取何值时，$f(z)$ 在复平面内处处解析？

\vspace{0.2cm}

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\item  %Problem 07
证明函数 $f(z)=e^x(\cos y + i\sin y)$ 在复平面上解析，且 $$f\,'(z)=f(z). $$

\vspace{0.2cm}

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\item  %Problem 08
设 $f(z)=u(x,y)+iv(x,y)$ 在区域 $D$ 内解析，并且 $v=u^2$, 求 $f(z)$. 

\vspace{0.2cm}

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\item  %Problem 09
设函数 $f(z)=u(x,y)+iv(x,y)$ 在区域 $D$ 内解析，且 $f\, '(z)\neq 0 \, (z\in D)$, 证明 
$$u(x,y)=c_1, v(x,y)=c_2,$$ 
其中 $c_1,c_2$ 为常数，是 $D$ 内两组正交曲线族。

\vspace{0.2cm}

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\item  %Problem 10
设 $u(x,y)$ 和 $v(x,y)$ 在区域 $D$ 内有一阶连续偏导数，则 $f(z)=u(x,y)+iv(x,y)$ 在 $D$ 内解析的充分必要条件为 $$\frac{\partial f}{\partial \bar{z}}(z) =0. $$

\vspace{0.2cm}


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\end{enumerate}


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\end{document}

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